Mathematical description of alloys solidification in a macro scale can be formulated using the one domain method (fixed domain approach). The energy equation corresponding to this model contains the parameter called a substitute thermal capacity (STC). The analytical form of STC results from the assumption concerning the course of the function fS = fS (T) describing the changes of solid state volumetric fraction and the temperature at the point considered. Between border temperatures TS , TL the function fS changes from 1 to 0. In this paper the volumetric fraction fS (more precisely fL = 1- fS ) is found using the simple models of macrosegregation (the lever arm rule, the Scheil model). In this way one obtains the formulas determining the course of STC resulting from the certain physical considerations and this approach seems to be closer to the real course of thermal processes proceeding in domain of solidifying alloy.
Thermal processes in domain of thin metal film subjected to a strong laser pulse are discussed. The heating of domain considered causes the melting and next (after the end of beam impact) the resolidification of metal superficial layer. The laser action (a time dependent belltype function) is taken into account by the introduction of internal heat source in the energy equation describing the heat transfer in domain of metal film. Taking into account the extremely short duration, extreme temperature gradients and very small geometrical dimensions of the domain considered, the mathematical model of the process is based on the dual phase lag equation supplemented by the suitable boundary-initial conditions. To model the phase transitions the artificial mushy zone is introduced. At the stage of numerical modeling the Control Volume Method is used. The examples of computations are also presented.
Heating process in the domain of thin metal film subjected to a strong laser pulse are discussed. The mathematical model of the process considered is based on the dual-phase-lag equation (DPLE) which results from the generalized form of the Fourier law. This approach is, first of all, used in the case of micro-scale heat transfer problems (the extremely short duration, extreme temperature gradients and very small geometrical dimensions of the domain considered). The external heating (a laser action) is substituted by the introduction of internal heat source to the DPLE. To model the melting process in domain of pure metal (chromium) the approach basing on the artificial mushy zone introduction is used and the main goal of investigation is the verification of influence of the artificial mushy zone ‘width’ on the results of melting modeling. At the stage of numerical modeling the author’s version of the Control Volume Method is used. In the final part of the paper the examples of computations and conclusions are presented.
In the paper the thermal processes proceeding in the solidifying metal are analyzed. The basic energy equation determining the course of solidification contains the component (source function) controlling the phase change. This component is proportional to the solidification rate ¶ fS/¶ t (fS Î [0, 1], is a temporary and local volumetric fraction of solid state). The value of fS can be found, among others, on the basic of laws determining the nucleation and nuclei growth. This approach leads to the so called micro/macro models (the second generation models). The capacity of internal heat source appearing in the equation concerning the macro scale (solidification and cooling of domain considered) results from the phenomena proceeding in the micro scale (nuclei growth). The function fS can be defined as a product of nuclei density N and single grain volume V (a linear model of crystallization) and this approach is applied in the paper presented. The problem discussed consists in the simultaneous identification of two parameters determining a course of solidification. In particular it is assumed that nuclei density N (micro scale) and volumetric specific heat of metal (macro scale) are unknown. Formulated in this way inverse problem is solved using the least squares criterion and gradient methods. The additional information which allows to identify the unknown parameters results from knowledge of cooling curves at the selected set of points from solidifying metal domain. On the stage of numerical realization the boundary element method is used. In the final part of the paper the examples of computations are presented.